My Homework 6
Bin Liu
2023-11-18
Importing data for Electric Vehicle Popultion Size History
library(readr)
Electric_Vehicle_Population_Size_History <- read_csv("~/Documents/Fall 2023 - Business Forecasting/Datasets/Electric_Vehicle_Population_Size_History.csv")## Rows: 80 Columns: 4
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ","
## chr (1): Date
## dbl (3): Plug-In Hybrid Electric Vehicle (PHEV) Count, Battery Electric Vehi...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.View(Electric_Vehicle_Population_Size_History)Creating a time series for Electric Vehicle (EV) Total
library(fpp)## Loading required package: forecast## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo## Loading required package: fma## Loading required package: expsmooth## Loading required package: lmtest## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric## Loading required package: tserieslibrary(fpp2)## ── Attaching packages ────────────────────────────────────────────── fpp2 2.5 ──## ✔ ggplot2 3.4.4## ##
## Attaching package: 'fpp2'## The following objects are masked from 'package:fpp':
##
## ausair, ausbeer, austa, austourists, debitcards, departures,
## elecequip, euretail, guinearice, oil, sunspotarea, usmeleclibrary(TTR)
Electric_Vehicle_Population_Total_ts <- ts(Electric_Vehicle_Population_Size_History$`Electric Vehicle (EV) Total`, frequency = 12, start = c(2017,1))
Electric_Vehicle_Population_Total_ts## Jan Feb Mar Apr May Jun Jul Aug Sep Oct
## 2017 22425 23094 23688 24390 25066 25737 26310 26825 27452 28173
## 2018 30267 30809 31573 32438 33634 34790 35633 36540 37454 39159
## 2019 43837 44579 45068 46344 47305 48385 49630 50583 51581 53436
## 2020 57137 57731 58606 58910 58915 59277 60813 61558 62338 63610
## 2021 68130 69286 70467 72790 74437 76145 78598 80537 82064 84200
## 2022 89954 91856 93908 96667 98524 100720 103480 106249 108061 111690
## 2023 121935 124930 128097 132365 136579 140832 144810 150482
## Nov Dec
## 2017 28765 29305
## 2018 40801 42065
## 2019 54060 54883
## 2020 65142 66230
## 2021 86245 88027
## 2022 115343 118127
## 2023attributes (Electric_Vehicle_Population_Total_ts)## $tsp
## [1] 2017.000 2023.583 12.000
##
## $class
## [1] "ts"plot(Electric_Vehicle_Population_Total_ts)
Acf(Electric_Vehicle_Population_Total_ts)
Forecasting Models
Average Model
The Average Model takes the average of all historical points. The plot shows that the avearge of the total registration of electric vehicles in the State of Washington is 66,273.58. This model will take this average to forecast for the next 24 months, which is the blue straight line in the plot. The shaded area shows the confidence interval that the future values may lie between this area. As we can see from the plot, this is not a good forecasting model for this data based on the trend since the forecasted values are way below the current trend.
mean_forecast <- meanf(Electric_Vehicle_Population_Total_ts, 24)
plot(mean_forecast)
Acf(mean_forecast$residuals)
attributes(mean_forecast)## $names
## [1] "method" "level" "x" "series" "mean" "lower"
## [7] "upper" "model" "lambda" "fitted" "residuals"
##
## $class
## [1] "forecast"summary(mean_forecast)##
## Forecast method: Mean
##
## Model Information:
## $mu
## [1] 66273.58
##
## $mu.se
## [1] 3832.713
##
## $sd
## [1] 34280.83
##
## $bootstrap
## [1] FALSE
##
## $call
## meanf(y = Electric_Vehicle_Population_Total_ts, h = 24)
##
## attr(,"class")
## [1] "meanf"
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set 7.275958e-12 34065.9 28119.51 -31.24979 56.11059 1.564725
## ACF1
## Training set 0.9498006
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Sep 2023 66273.58 21694.38 110852.8 -2385.846 134933
## Oct 2023 66273.58 21694.38 110852.8 -2385.846 134933
## Nov 2023 66273.58 21694.38 110852.8 -2385.846 134933
## Dec 2023 66273.58 21694.38 110852.8 -2385.846 134933
## Jan 2024 66273.58 21694.38 110852.8 -2385.846 134933
## Feb 2024 66273.58 21694.38 110852.8 -2385.846 134933
## Mar 2024 66273.58 21694.38 110852.8 -2385.846 134933
## Apr 2024 66273.58 21694.38 110852.8 -2385.846 134933
## May 2024 66273.58 21694.38 110852.8 -2385.846 134933
## Jun 2024 66273.58 21694.38 110852.8 -2385.846 134933
## Jul 2024 66273.58 21694.38 110852.8 -2385.846 134933
## Aug 2024 66273.58 21694.38 110852.8 -2385.846 134933
## Sep 2024 66273.58 21694.38 110852.8 -2385.846 134933
## Oct 2024 66273.58 21694.38 110852.8 -2385.846 134933
## Nov 2024 66273.58 21694.38 110852.8 -2385.846 134933
## Dec 2024 66273.58 21694.38 110852.8 -2385.846 134933
## Jan 2025 66273.58 21694.38 110852.8 -2385.846 134933
## Feb 2025 66273.58 21694.38 110852.8 -2385.846 134933
## Mar 2025 66273.58 21694.38 110852.8 -2385.846 134933
## Apr 2025 66273.58 21694.38 110852.8 -2385.846 134933
## May 2025 66273.58 21694.38 110852.8 -2385.846 134933
## Jun 2025 66273.58 21694.38 110852.8 -2385.846 134933
## Jul 2025 66273.58 21694.38 110852.8 -2385.846 134933
## Aug 2025 66273.58 21694.38 110852.8 -2385.846 134933Naive Model
The Naive Model takes the last month’s data to forecast for the next 24 months as shown in the blue straight line in the plot. The confidence interval shows the future values may lie within this shaded area based on this prediction. This forecasting model does not appear to be a good model based on the trend.
naive_forecast <- naive(Electric_Vehicle_Population_Total_ts, 24)
plot(naive_forecast)
Acf(naive_forecast$residuals)
summary(naive_forecast)##
## Forecast method: Naive method
##
## Model Information:
## Call: naive(y = Electric_Vehicle_Population_Total_ts, h = 24)
##
## Residual sd: 1971.8019
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 1620.975 1971.802 1620.975 2.377577 2.377577 0.09019999 0.7573396
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Sep 2023 150482 147955.0 153009.0 146617.3 154346.7
## Oct 2023 150482 146908.3 154055.7 145016.5 155947.5
## Nov 2023 150482 146105.2 154858.8 143788.2 157175.8
## Dec 2023 150482 145428.1 155535.9 142752.7 158211.3
## Jan 2024 150482 144831.5 156132.5 141840.4 159123.6
## Feb 2024 150482 144292.2 156671.8 141015.6 159948.4
## Mar 2024 150482 143796.3 157167.7 140257.1 160706.9
## Apr 2024 150482 143334.7 157629.3 139551.1 161412.9
## May 2024 150482 142901.1 158062.9 138888.0 162076.0
## Jun 2024 150482 142491.0 158473.0 138260.9 162703.1
## Jul 2024 150482 142101.0 158863.0 137664.4 163299.6
## Aug 2024 150482 141728.3 159235.7 137094.4 163869.6
## Sep 2024 150482 141370.9 159593.1 136547.8 164416.2
## Oct 2024 150482 141027.0 159937.0 136021.8 164942.2
## Nov 2024 150482 140695.1 160268.9 135514.2 165449.8
## Dec 2024 150482 140374.1 160589.9 135023.4 165940.6
## Jan 2025 150482 140063.1 160900.9 134547.6 166416.4
## Feb 2025 150482 139761.0 161203.0 134085.6 166878.4
## Mar 2025 150482 139467.2 161496.8 133636.3 167327.7
## Apr 2025 150482 139181.1 161782.9 133198.7 167765.3
## May 2025 150482 138902.0 162062.0 132771.9 168192.1
## Jun 2025 150482 138629.5 162334.5 132355.1 168608.9
## Jul 2025 150482 138363.1 162600.9 131947.7 169016.3
## Aug 2025 150482 138102.4 162861.6 131549.1 169414.9Random Walk Forecast (RWF)
Similar to the Naive Model, the Random Walk Forecast Model also takes the last month’s data to forecast for the values in the next 24 months with the same values for the confidence interval. Again, it is not considered a good forecasting model based on the trend.
rwf_forecast <- rwf(Electric_Vehicle_Population_Total_ts,24)
plot(rwf_forecast)
Acf(rwf_forecast$residuals)
summary(rwf_forecast)##
## Forecast method: Random walk
##
## Model Information:
## Call: rwf(y = Electric_Vehicle_Population_Total_ts, h = 24)
##
## Residual sd: 1971.8019
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 1620.975 1971.802 1620.975 2.377577 2.377577 0.09019999 0.7573396
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Sep 2023 150482 147955.0 153009.0 146617.3 154346.7
## Oct 2023 150482 146908.3 154055.7 145016.5 155947.5
## Nov 2023 150482 146105.2 154858.8 143788.2 157175.8
## Dec 2023 150482 145428.1 155535.9 142752.7 158211.3
## Jan 2024 150482 144831.5 156132.5 141840.4 159123.6
## Feb 2024 150482 144292.2 156671.8 141015.6 159948.4
## Mar 2024 150482 143796.3 157167.7 140257.1 160706.9
## Apr 2024 150482 143334.7 157629.3 139551.1 161412.9
## May 2024 150482 142901.1 158062.9 138888.0 162076.0
## Jun 2024 150482 142491.0 158473.0 138260.9 162703.1
## Jul 2024 150482 142101.0 158863.0 137664.4 163299.6
## Aug 2024 150482 141728.3 159235.7 137094.4 163869.6
## Sep 2024 150482 141370.9 159593.1 136547.8 164416.2
## Oct 2024 150482 141027.0 159937.0 136021.8 164942.2
## Nov 2024 150482 140695.1 160268.9 135514.2 165449.8
## Dec 2024 150482 140374.1 160589.9 135023.4 165940.6
## Jan 2025 150482 140063.1 160900.9 134547.6 166416.4
## Feb 2025 150482 139761.0 161203.0 134085.6 166878.4
## Mar 2025 150482 139467.2 161496.8 133636.3 167327.7
## Apr 2025 150482 139181.1 161782.9 133198.7 167765.3
## May 2025 150482 138902.0 162062.0 132771.9 168192.1
## Jun 2025 150482 138629.5 162334.5 132355.1 168608.9
## Jul 2025 150482 138363.1 162600.9 131947.7 169016.3
## Aug 2025 150482 138102.4 162861.6 131549.1 169414.9A drift with RWF
The Random Walk Forecast Model with a drift shows an upward trend for the next 24 months. The confidence interval also has an increased values as opposed to the Random Walk Forecast Model. This shows a better forecast based on the trend of the time series.
rwf_forecast_drift <- rwf(Electric_Vehicle_Population_Total_ts,24, drift=TRUE)
plot(rwf_forecast_drift)
Acf(rwf_forecast_drift$residuals)
summary(rwf_forecast_drift)##
## Forecast method: Random walk with drift
##
## Model Information:
## Call: rwf(y = Electric_Vehicle_Population_Total_ts, h = 24, drift = TRUE)
##
## Drift: 1620.9747 (se 127.1202)
## Residual sd: 1129.8687
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE
## Training set -1.842015e-13 1122.695 873.4901 -0.7817773 1.556505 0.04860582
## ACF1
## Training set 0.7573396
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Sep 2023 152103.0 150645.9 153560.1 149874.5 154331.4
## Oct 2023 153723.9 151650.4 155797.5 150552.8 156895.1
## Nov 2023 155344.9 152789.8 157900.1 151437.1 159252.7
## Dec 2023 156965.9 153997.5 159934.3 152426.2 161505.6
## Jan 2024 158586.9 155248.2 161925.6 153480.8 163692.9
## Feb 2024 160207.8 156528.8 163886.9 154581.2 165834.5
## Mar 2024 161828.8 157831.7 165826.0 155715.7 167941.9
## Apr 2024 163449.8 159151.9 167747.7 156876.7 170022.9
## May 2024 165070.8 160486.0 169655.5 158059.0 172082.5
## Jun 2024 166691.7 161831.6 171551.9 159258.9 174124.6
## Jul 2024 168312.7 163186.8 173438.6 160473.4 176152.1
## Aug 2024 169933.7 164550.2 175317.2 161700.4 178167.0
## Sep 2024 171554.7 165920.7 177188.7 162938.2 180171.1
## Oct 2024 173175.6 167297.3 179054.0 164185.5 182165.8
## Nov 2024 174796.6 168679.3 180913.9 165441.0 184152.2
## Dec 2024 176417.6 170066.2 182769.0 166703.9 186131.3
## Jan 2025 178038.6 171457.3 184619.9 167973.4 188103.8
## Feb 2025 179659.5 172852.3 186466.8 169248.7 190070.4
## Mar 2025 181280.5 174250.8 188310.3 170529.4 192031.6
## Apr 2025 182901.5 175652.4 190150.6 171815.0 193988.0
## May 2025 184522.5 177056.9 191988.0 173104.9 195940.0
## Jun 2025 186143.4 178464.1 193822.8 174398.9 197887.9
## Jul 2025 187764.4 179873.7 195655.1 175696.7 199832.2
## Aug 2025 189385.4 181285.6 197485.2 176997.8 201773.0Seasonal Naive
The time series shows an upward trend and does not show any seasonality trend. Therefore, the Seasnoal Naive Model is not a good model. Based on the plot, it appears that it takes the upward trend from the recent months then a downward trend, then a similar upward trend again to forecast the values for the next 24 months. This model does not represent the increasing upward trend of the time series.
snaive_forecast <- snaive(Electric_Vehicle_Population_Total_ts,24)
plot(snaive_forecast)
Acf(snaive_forecast$residuals)
summary(snaive_forecast)##
## Forecast method: Seasonal naive method
##
## Model Information:
## Call: snaive(y = Electric_Vehicle_Population_Total_ts, h = 24)
##
## Residual sd: 20142.9651
##
## Error measures:
## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 17970.9 20142.97 17970.9 24.40798 24.40798 1 0.9243048
##
## Forecasts:
## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Sep 2023 108061 82246.75 133875.2 68581.51 147540.5
## Oct 2023 111690 85875.75 137504.2 72210.51 151169.5
## Nov 2023 115343 89528.75 141157.2 75863.51 154822.5
## Dec 2023 118127 92312.75 143941.2 78647.51 157606.5
## Jan 2024 121935 96120.75 147749.2 82455.51 161414.5
## Feb 2024 124930 99115.75 150744.2 85450.51 164409.5
## Mar 2024 128097 102282.75 153911.2 88617.51 167576.5
## Apr 2024 132365 106550.75 158179.2 92885.51 171844.5
## May 2024 136579 110764.75 162393.2 97099.51 176058.5
## Jun 2024 140832 115017.75 166646.2 101352.51 180311.5
## Jul 2024 144810 118995.75 170624.2 105330.51 184289.5
## Aug 2024 150482 124667.75 176296.2 111002.51 189961.5
## Sep 2024 108061 71554.14 144567.9 52228.58 163893.4
## Oct 2024 111690 75183.14 148196.9 55857.58 167522.4
## Nov 2024 115343 78836.14 151849.9 59510.58 171175.4
## Dec 2024 118127 81620.14 154633.9 62294.58 173959.4
## Jan 2025 121935 85428.14 158441.9 66102.58 177767.4
## Feb 2025 124930 88423.14 161436.9 69097.58 180762.4
## Mar 2025 128097 91590.14 164603.9 72264.58 183929.4
## Apr 2025 132365 95858.14 168871.9 76532.58 188197.4
## May 2025 136579 100072.14 173085.9 80746.58 192411.4
## Jun 2025 140832 104325.14 177338.9 84999.58 196664.4
## Jul 2025 144810 108303.14 181316.9 88977.58 200642.4
## Aug 2025 150482 113975.14 186988.9 94649.58 206314.4Moving Averages
Moving average with order of 3
Based on the ACF, it shows that the recent data are more significant than the historical data. Therefore, the lower order moving average model provides a better forecast than higher order moving average model since the larger weight is given to the more recent period.
MA3_forecast <- ma(Electric_Vehicle_Population_Total_ts,order=3)
plot(MA3_forecast)
Moving average with order of 6
The moving average model with order of 6 still provides a good forecast for the upward trend although it smooths out the data more than the previous model with order of 3. It gives more weight to more historical data which appears to be less significant. This is not as good as the previous model with the lower order.
MA6_forecast <- ma(Electric_Vehicle_Population_Total_ts,order=6)
plot(MA6_forecast)
Moving average with order of 12
The larger order of the moving average model, the smoother effect on the plot. As we can see from the plot, it gives more weight to more historical data as comparing to the moving average model with order of 3 and 6. Therefore, it is not as good as the previous two models with the lower order.
MA12_forecast <- ma(Electric_Vehicle_Population_Total_ts,order=12)
plot(MA12_forecast)
Moving average with order of 18
As mentioned above, the larger order of moving average puts more weight on more historical data. Therefore, the moving average with large order does not represent well for this time series.
MA18_forecast <- ma(Electric_Vehicle_Population_Total_ts,order=18)
plot(MA18_forecast)
Exponential
ets_forecast <- ets(Electric_Vehicle_Population_Total_ts)
plot(ets_forecast)
Holtwinters
The Holtwinters Model without any smoothing constants provides a pretty good fit to the original data plot. The residuls are very small based on the comparison of the actual and forecasted data (black line vs red line).
HW_forecast <- HoltWinters(Electric_Vehicle_Population_Total_ts)
plot(HW_forecast)
attributes(HW_forecast)## $names
## [1] "fitted" "x" "alpha" "beta" "gamma"
## [6] "coefficients" "seasonal" "SSE" "call"
##
## $class
## [1] "HoltWinters"summary(HW_forecast)## Length Class Mode
## fitted 272 mts numeric
## x 80 ts numeric
## alpha 1 -none- numeric
## beta 1 -none- numeric
## gamma 1 -none- numeric
## coefficients 14 -none- numeric
## seasonal 1 -none- character
## SSE 1 -none- numeric
## call 2 -none- callplot(HW_forecast)
The Holtwinters Model with exponential smoothing for trend and non-seasonal model provides a decent fit. However, the residual are greater than the previous model with no smoothing constants.
SSE_Simple <- HoltWinters(Electric_Vehicle_Population_Total_ts,beta=FALSE,gamma=FALSE)
attributes(SSE_Simple)## $names
## [1] "fitted" "x" "alpha" "beta" "gamma"
## [6] "coefficients" "seasonal" "SSE" "call"
##
## $class
## [1] "HoltWinters"plot(SSE_Simple)
SSE_Simple$SSE## [1] 307178580head(SSE_Simple$fitted)## xhat level
## Feb 2017 22425.00 22425.00
## Mar 2017 23093.97 23093.97
## Apr 2017 23687.97 23687.97
## May 2017 24389.97 24389.97
## Jun 2017 25065.97 25065.97
## Jul 2017 25736.97 25736.97The Holtwinters Model with non-exponential smoothing for trend and non-seasonal model provides a pretty nice fit and the residuals are small among the acutal and forecasted data.
SSE_Simple_1 <- HoltWinters(Electric_Vehicle_Population_Total_ts,beta=TRUE,gamma=FALSE)
attributes(SSE_Simple_1)## $names
## [1] "fitted" "x" "alpha" "beta" "gamma"
## [6] "coefficients" "seasonal" "SSE" "call"
##
## $class
## [1] "HoltWinters"plot(SSE_Simple_1)
SSE_Simple_1$SSE## [1] 22118991head(SSE_Simple_1$fitted)## xhat level trend
## Mar 2017 23763.00 23094.00 669.0000
## Apr 2017 24342.79 23718.40 624.3968
## May 2017 25023.34 24370.87 652.4709
## Jun 2017 25726.55 25048.71 677.8421
## Jul 2017 26416.82 25732.77 684.0557
## Aug 2017 26973.82 26353.29 620.5281The Holtwinters Model with non-exponential smoothing for trend and seasonal model also provides a pretty nice fit and the residuals are small among the acutal and forecasted data.
SSE_Simple_2 <- HoltWinters(Electric_Vehicle_Population_Total_ts,beta=TRUE,gamma=TRUE)
attributes(SSE_Simple_2)## $names
## [1] "fitted" "x" "alpha" "beta" "gamma"
## [6] "coefficients" "seasonal" "SSE" "call"
##
## $class
## [1] "HoltWinters"plot(SSE_Simple_2)
SSE_Simple_2$SSE## [1] 56839993head(SSE_Simple_2$fitted)## xhat level trend season
## Jan 2018 26034.51 25199.69 775.1234 59.69792
## Feb 2018 34283.26 29837.25 4637.5633 -191.55208
## Mar 2018 32522.28 31304.31 1467.0597 -249.09375
## Apr 2018 32247.28 31905.09 600.7795 -258.59375
## May 2018 33432.90 32679.92 774.8285 -21.84375
## Jun 2018 34697.59 33638.26 958.3431 100.98958The Holtwinters Model with exponential smoothing for trend and seasonal model provides quite similar forecast with the exponential smoothing for trend and non-seaonal model. In both models, the residuals are greater than the other Holtwinters models displayed above.
SSE_Simple_3 <- HoltWinters(Electric_Vehicle_Population_Total_ts,beta=FALSE,gamma=TRUE)
attributes(SSE_Simple_3)## $names
## [1] "fitted" "x" "alpha" "beta" "gamma"
## [6] "coefficients" "seasonal" "SSE" "call"
##
## $class
## [1] "HoltWinters"plot(SSE_Simple_3)
SSE_Simple_3$SSE## [1] 325250864head(SSE_Simple_2$fitted)## xhat level trend season
## Jan 2018 26034.51 25199.69 775.1234 59.69792
## Feb 2018 34283.26 29837.25 4637.5633 -191.55208
## Mar 2018 32522.28 31304.31 1467.0597 -249.09375
## Apr 2018 32247.28 31905.09 600.7795 -258.59375
## May 2018 33432.90 32679.92 774.8285 -21.84375
## Jun 2018 34697.59 33638.26 958.3431 100.98958Plot all in a single chart
plot(mean_forecast, col="black")
lines(naive_forecast$mean,col="red")
lines(rwf_forecast$mean, col="green")
lines(rwf_forecast_drift$mean, col="hotpink")
lines(snaive_forecast$mean, col="blue")
lines(MA6_forecast, col="purple")
lines(MA12_forecast, col="orange")
lines(MA18_forecast, col="yellow")
Decomposition
ets_forecast <- ets(Electric_Vehicle_Population_Total_ts)
plot(ets_forecast)
attributes(ets_forecast)## $names
## [1] "loglik" "aic" "bic" "aicc" "mse"
## [6] "amse" "fit" "residuals" "fitted" "states"
## [11] "par" "m" "method" "series" "components"
## [16] "call" "initstate" "sigma2" "x"
##
## $class
## [1] "ets"ets_forecast$mse## [1] 275304.6Accuracy measures for different forecasting models
I pick the Mean Absolute Percentage Error (MAPE) as the accuracy measure for this time series. By comparing the MAPE with different models listed below, the Moving Average with order of 3 shows the least MAPE. This indicates that the Moving Average with order of 3 is the best model based on the accuracy comparison.
library(forecast)Average Model
accuracy(mean_forecast)## ME RMSE MAE MPE MAPE MASE
## Training set 7.275958e-12 34065.9 28119.51 -31.24979 56.11059 1.564725
## ACF1
## Training set 0.9498006Naive Model
accuracy(naive_forecast)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 1620.975 1971.802 1620.975 2.377577 2.377577 0.09019999 0.7573396Random Walk Forecast Model
accuracy(rwf_forecast)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 1620.975 1971.802 1620.975 2.377577 2.377577 0.09019999 0.7573396Random Walk Forecast with a Drift Model
accuracy(rwf_forecast_drift)## ME RMSE MAE MPE MAPE MASE
## Training set -1.842015e-13 1122.695 873.4901 -0.7817773 1.556505 0.04860582
## ACF1
## Training set 0.7573396Seasonal Naive Model
accuracy(snaive_forecast)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 17970.9 20142.97 17970.9 24.40798 24.40798 1 0.9243048Moving Averages with different orders (3, 6, 12 and 18)
accuracy(MA3_forecast,Electric_Vehicle_Population_Total_ts, h=24)## ME RMSE MAE MPE MAPE ACF1 Theil's U
## Test set -21.38034 210.1715 150.765 -0.02526429 0.2403089 -0.414049 0.1283632accuracy(MA6_forecast,Electric_Vehicle_Population_Total_ts, h=24)## ME RMSE MAE MPE MAPE ACF1 Theil's U
## Test set -76.85473 308.7977 232.2872 -0.1015902 0.3866858 0.1726912 0.1968508accuracy(MA12_forecast,Electric_Vehicle_Population_Total_ts, h=24)## ME RMSE MAE MPE MAPE ACF1 Theil's U
## Test set -267.8989 575.2414 459.6562 -0.3930565 0.7614215 0.6495116 0.3453847accuracy(MA18_forecast,Electric_Vehicle_Population_Total_ts, h=24)## ME RMSE MAE MPE MAPE ACF1 Theil's U
## Test set -566.4534 997.2238 827.0959 -0.8592732 1.360718 0.8454964 0.5512701Holtwinters with different combinations of smoothing trends and seasonal models
forecast_ets_hw <- forecast(HW_forecast, h=24)
accuracy(forecast_ets_hw)## ME RMSE MAE MPE MAPE MASE
## Training set 166.8618 774.4541 502.284 0.1987327 0.8978803 0.02794986
## ACF1
## Training set 0.00181798forecast_ets_SSE_Simple <- forecast(SSE_Simple, h=24)
accuracy(forecast_ets_SSE_Simple)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 1621.049 1971.887 1621.049 2.377686 2.377686 0.09020412 0.7573582forecast_ets_SSE_Simple_1 <- forecast(SSE_Simple_1, h=24)
accuracy(forecast_ets_SSE_Simple_1)## ME RMSE MAE MPE MAPE MASE
## Training set 93.72343 532.5193 404.7442 0.1066907 0.6409175 0.0225222
## ACF1
## Training set 0.03504061forecast_ets_SSE_Simple_2 <- forecast(SSE_Simple_2, h=24)
accuracy(forecast_ets_SSE_Simple_2)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 80.08842 914.266 590.824 0.08649111 1.079665 0.03287671 -0.3162797forecast_ets_SSE_Simple_3 <- forecast(SSE_Simple_3, h=24)
accuracy(forecast_ets_SSE_Simple_3)## ME RMSE MAE MPE MAPE MASE ACF1
## Training set 1840.69 2187.03 1847.504 2.572231 2.583797 0.1028054 0.6318507Seasonal Decomposition of Time Series by Loess
stl_decomp <- stl(Electric_Vehicle_Population_Total_ts, s.window = "periodic")
plot(stl_decomp)
attributes (stl_decomp)## $names
## [1] "time.series" "weights" "call" "win" "deg"
## [6] "jump" "inner" "outer"
##
## $class
## [1] "stl"stl_decomp$weights## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [39] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
## [77] 1 1 1 1stl_decomp$call## stl(x = Electric_Vehicle_Population_Total_ts, s.window = "periodic")stl_decomp$win## s t l
## 801 19 13stl_decomp$deg## s t l
## 0 1 1stl_decomp$jump## s t l
## 81 2 2stl_decomp$inner## [1] 2stl_decomp$outer## [1] 0Seasonal adjustment
tmp <- seasadj(stl_decomp)
plot(Electric_Vehicle_Population_Total_ts, col="black")
lines(tmp, col="red")
Forecast of Seasonal Decomposition of Time Series by Loess
f_stl <- forecast(stl_decomp)
f_stl <- forecast(stl_decomp, h=24)
plot(f_stl)
accuracy(f_stl)## ME RMSE MAE MPE MAPE MASE
## Training set 98.78592 537.7358 378.9391 0.1096643 0.6958745 0.02108627
## ACF1
## Training set -0.02587675attributes(f_stl)## $names
## [1] "model" "mean" "level" "x" "upper" "lower"
## [7] "fitted" "method" "series" "residuals"
##
## $class
## [1] "forecast"Acf(f_stl$residuals)
Classical Seasonal Decomposition by Moving Averages
decomp_ev <- decompose(Electric_Vehicle_Population_Total_ts)
plot(decomp_ev)
attributes (decomp_ev)## $names
## [1] "x" "seasonal" "trend" "random" "figure" "type"
##
## $class
## [1] "decomposed.ts"Classical Seasonal Decomposition by Moving Averages - Multiplicative
decomp_evm <- decompose(Electric_Vehicle_Population_Total_ts, type="multiplicative")
plot(decomp_evm)
attributes (decomp_evm)## $names
## [1] "x" "seasonal" "trend" "random" "figure" "type"
##
## $class
## [1] "decomposed.ts"Forecast of Classical Seasonal Decomposition by Moving Averages
By using two different seasonal decomposition methods, the forecasting accuracy measure for Classical Seasonal Decomposition by Moving Averages has lower MAPE. The ACF plot for residuals also looks better for Classical Seasonal Decomposition by Moving Averages as the lines are all within the blue dash lines on the plot.
f_decomp <- forecast(seasadj(decomp_ev), h=24)
plot(f_decomp)
accuracy(f_decomp)## ME RMSE MAE MPE MAPE MASE
## Training set 97.10534 438.1027 329.2923 0.09248334 0.5842102 0.01832364
## ACF1
## Training set -0.01195674Acf(f_decomp$residuals)
ARIMA Model
ADF test
The p-value for ADF test is 0.99 which is greater than 0.05. This indicates that the differencing is required based on ADF test.
adf.test(Electric_Vehicle_Population_Total_ts)## Warning in adf.test(Electric_Vehicle_Population_Total_ts): p-value greater than
## printed p-value##
## Augmented Dickey-Fuller Test
##
## data: Electric_Vehicle_Population_Total_ts
## Dickey-Fuller = 2.9404, Lag order = 4, p-value = 0.99
## alternative hypothesis: stationaryKipps test
The p-value for Kipps test is 0.01 which is less than 0.05. This indicates that the differencing is required based on Kipps test.
kpss.test(Electric_Vehicle_Population_Total_ts)## Warning in kpss.test(Electric_Vehicle_Population_Total_ts): p-value smaller
## than printed p-value##
## KPSS Test for Level Stationarity
##
## data: Electric_Vehicle_Population_Total_ts
## KPSS Level = 1.986, Truncation lag parameter = 3, p-value = 0.01NSDIFFS for seasonal data
NSDIFFS works for seasonal data. Since this time series doesn’t show any seasonality, therefore, the result turns out to be 0 based on NSDIFFS function.
nsdiffs(Electric_Vehicle_Population_Total_ts)## [1] 0NDIFFS for non-seasonal data
NDIFFS function shows that this time series requires 2 differences to become stationary.
ndiffs(Electric_Vehicle_Population_Total_ts)## [1] 2tsdisplay plots ACF, PACF and timeseries plot together
tsdisplay(Electric_Vehicle_Population_Total_ts)
Take first order differences as indiciated by NDIFFS function
Electric_Vehicle_Population_Total_ts_diff1 <- diff(Electric_Vehicle_Population_Total_ts, differences = 1)See if more differences is needed and how plots look
The plot for 1st order difference shows that the time series becomes quite stationary from 2017 to 2021 then slight upward trending after year 2021. Based on ACF, the historical data becomes less and less significant. Based on PACF, the first number for P may 1, 2, or 3 since the first, second and third lags are above the range line.
tsdisplay(Electric_Vehicle_Population_Total_ts_diff1)
ndiffs(Electric_Vehicle_Population_Total_ts_diff1)## [1] 1kpss.test(Electric_Vehicle_Population_Total_ts_diff1)## Warning in kpss.test(Electric_Vehicle_Population_Total_ts_diff1): p-value
## smaller than printed p-value##
## KPSS Test for Level Stationarity
##
## data: Electric_Vehicle_Population_Total_ts_diff1
## KPSS Level = 1.5831, Truncation lag parameter = 3, p-value = 0.01adf.test(Electric_Vehicle_Population_Total_ts_diff1)## Warning in adf.test(Electric_Vehicle_Population_Total_ts_diff1): p-value
## greater than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: Electric_Vehicle_Population_Total_ts_diff1
## Dickey-Fuller = 0.11096, Lag order = 4, p-value = 0.99
## alternative hypothesis: stationaryTake second order differences as indicated by NDIFFS function
Based on the 2nd order of differencing, the time series is now stationary. The ACF has positive and negative values which implies the change of direction of the time series. The PACF still shows that the first and second lags are signification. The values for P may fall within 1 or 2.
Electric_Vehicle_Population_Total_ts_diff2 <- diff(Electric_Vehicle_Population_Total_ts, differences = 2)
tsdisplay(Electric_Vehicle_Population_Total_ts_diff2)
ndiffs(Electric_Vehicle_Population_Total_ts_diff2)## [1] 0kpss.test(Electric_Vehicle_Population_Total_ts_diff2)## Warning in kpss.test(Electric_Vehicle_Population_Total_ts_diff2): p-value
## greater than printed p-value##
## KPSS Test for Level Stationarity
##
## data: Electric_Vehicle_Population_Total_ts_diff2
## KPSS Level = 0.30392, Truncation lag parameter = 3, p-value = 0.1adf.test(Electric_Vehicle_Population_Total_ts_diff2)## Warning in adf.test(Electric_Vehicle_Population_Total_ts_diff2): p-value
## smaller than printed p-value##
## Augmented Dickey-Fuller Test
##
## data: Electric_Vehicle_Population_Total_ts_diff2
## Dickey-Fuller = -5.6299, Lag order = 4, p-value = 0.01
## alternative hypothesis: stationaryWe now know that the d part in ARIMA (p,d,q) is 2. Now let’s find p and q
acf(Electric_Vehicle_Population_Total_ts_diff2, lag.max=20)
Acf(Electric_Vehicle_Population_Total_ts_diff2, lag.max=20)
Plot a correlogram
Acf(Electric_Vehicle_Population_Total_ts_diff2, lag.max=20)
Acf(Electric_Vehicle_Population_Total_ts_diff2, lag.max=20, plot=FALSE)##
## Autocorrelations of series 'Electric_Vehicle_Population_Total_ts_diff2', by lag
##
## 0 1 2 3 4 5 6 7 8 9 10
## 1.000 -0.414 -0.238 0.408 -0.176 -0.149 0.340 -0.125 -0.250 0.362 -0.059
## 11 12 13 14 15 16 17 18 19 20
## -0.295 0.386 -0.115 -0.202 0.313 -0.095 -0.215 0.270 -0.028 -0.285Plot a partial correlogram
Pacf(Electric_Vehicle_Population_Total_ts_diff2, lag.max=20)
Pacf(Electric_Vehicle_Population_Total_ts_diff2, lag.max=20, plot=FALSE)##
## Partial autocorrelations of series 'Electric_Vehicle_Population_Total_ts_diff2', by lag
##
## 1 2 3 4 5 6 7 8 9 10 11
## -0.414 -0.494 0.090 -0.013 -0.088 0.176 0.131 -0.145 0.082 0.115 -0.113
## 12 13 14 15 16 17 18 19 20
## 0.121 0.018 0.010 0.075 0.017 -0.050 -0.006 0.006 -0.132Auto ARIMA, BIC and AIC
Based on auto.arima function, the best model is ARIMA (2,2,0) (1,0,0) [12]. The AIC value is 1191.86 and BIC value is 1201.29.
auto_fit <- auto.arima(Electric_Vehicle_Population_Total_ts, approximation = FALSE, stepwise = FALSE, trace = TRUE)##
## ARIMA(0,2,0) : 1229.067
## ARIMA(0,2,0)(0,0,1)[12] : 1220.337
## ARIMA(0,2,0)(0,0,2)[12] : 1216.092
## ARIMA(0,2,0)(1,0,0)[12] : 1214.323
## ARIMA(0,2,0)(1,0,1)[12] : Inf
## ARIMA(0,2,0)(1,0,2)[12] : Inf
## ARIMA(0,2,0)(2,0,0)[12] : 1214.612
## ARIMA(0,2,0)(2,0,1)[12] : Inf
## ARIMA(0,2,0)(2,0,2)[12] : Inf
## ARIMA(0,2,1) : 1205.252
## ARIMA(0,2,1)(0,0,1)[12] : 1199.25
## ARIMA(0,2,1)(0,0,2)[12] : 1194.864
## ARIMA(0,2,1)(1,0,0)[12] : 1194.567
## ARIMA(0,2,1)(1,0,1)[12] : 1194.865
## ARIMA(0,2,1)(1,0,2)[12] : 1197.103
## ARIMA(0,2,1)(2,0,0)[12] : 1194.505
## ARIMA(0,2,1)(2,0,1)[12] : 1196.38
## ARIMA(0,2,1)(2,0,2)[12] : 1198.69
## ARIMA(0,2,2) : 1204.55
## ARIMA(0,2,2)(0,0,1)[12] : 1200.12
## ARIMA(0,2,2)(0,0,2)[12] : 1196.339
## ARIMA(0,2,2)(1,0,0)[12] : 1196.18
## ARIMA(0,2,2)(1,0,1)[12] : 1196.59
## ARIMA(0,2,2)(1,0,2)[12] : 1198.92
## ARIMA(0,2,2)(2,0,0)[12] : 1196.192
## ARIMA(0,2,2)(2,0,1)[12] : 1197.983
## ARIMA(0,2,3) : 1201.069
## ARIMA(0,2,3)(0,0,1)[12] : 1197.845
## ARIMA(0,2,3)(0,0,2)[12] : 1196.766
## ARIMA(0,2,3)(1,0,0)[12] : 1195.469
## ARIMA(0,2,3)(1,0,1)[12] : 1196.219
## ARIMA(0,2,3)(2,0,0)[12] : 1196.315
## ARIMA(0,2,4) : 1202.199
## ARIMA(0,2,4)(0,0,1)[12] : 1199.753
## ARIMA(0,2,4)(1,0,0)[12] : 1197.688
## ARIMA(0,2,5) : 1203.513
## ARIMA(1,2,0) : 1216.036
## ARIMA(1,2,0)(0,0,1)[12] : 1208.277
## ARIMA(1,2,0)(0,0,2)[12] : 1202.341
## ARIMA(1,2,0)(1,0,0)[12] : 1202.311
## ARIMA(1,2,0)(1,0,1)[12] : 1202.658
## ARIMA(1,2,0)(1,0,2)[12] : Inf
## ARIMA(1,2,0)(2,0,0)[12] : 1202.186
## ARIMA(1,2,0)(2,0,1)[12] : 1203.704
## ARIMA(1,2,0)(2,0,2)[12] : 1206.044
## ARIMA(1,2,1) : 1206.448
## ARIMA(1,2,1)(0,0,1)[12] : 1200.93
## ARIMA(1,2,1)(0,0,2)[12] : 1196.666
## ARIMA(1,2,1)(1,0,0)[12] : 1196.476
## ARIMA(1,2,1)(1,0,1)[12] : 1196.852
## ARIMA(1,2,1)(1,0,2)[12] : Inf
## ARIMA(1,2,1)(2,0,0)[12] : 1196.451
## ARIMA(1,2,1)(2,0,1)[12] : 1198.294
## ARIMA(1,2,2) : 1203.532
## ARIMA(1,2,2)(0,0,1)[12] : 1199.261
## ARIMA(1,2,2)(0,0,2)[12] : 1197.133
## ARIMA(1,2,2)(1,0,0)[12] : 1196.038
## ARIMA(1,2,2)(1,0,1)[12] : Inf
## ARIMA(1,2,2)(2,0,0)[12] : Inf
## ARIMA(1,2,3) : 1202.229
## ARIMA(1,2,3)(0,0,1)[12] : 1199.598
## ARIMA(1,2,3)(1,0,0)[12] : 1197.536
## ARIMA(1,2,4) : 1204.382
## ARIMA(2,2,0) : 1195.062
## ARIMA(2,2,0)(0,0,1)[12] : 1193.853
## ARIMA(2,2,0)(0,0,2)[12] : 1193.337
## ARIMA(2,2,0)(1,0,0)[12] : 1192.409
## ARIMA(2,2,0)(1,0,1)[12] : 1193.507
## ARIMA(2,2,0)(1,0,2)[12] : Inf
## ARIMA(2,2,0)(2,0,0)[12] : 1193.398
## ARIMA(2,2,0)(2,0,1)[12] : 1195.189
## ARIMA(2,2,1) : 1196.847
## ARIMA(2,2,1)(0,0,1)[12] : 1195.955
## ARIMA(2,2,1)(0,0,2)[12] : 1195.672
## ARIMA(2,2,1)(1,0,0)[12] : 1194.654
## ARIMA(2,2,1)(1,0,1)[12] : 1195.854
## ARIMA(2,2,1)(2,0,0)[12] : 1195.748
## ARIMA(2,2,2) : 1198.8
## ARIMA(2,2,2)(0,0,1)[12] : 1198.274
## ARIMA(2,2,2)(1,0,0)[12] : 1197
## ARIMA(2,2,3) : Inf
## ARIMA(3,2,0) : 1196.761
## ARIMA(3,2,0)(0,0,1)[12] : 1195.946
## ARIMA(3,2,0)(0,0,2)[12] : 1195.673
## ARIMA(3,2,0)(1,0,0)[12] : 1194.657
## ARIMA(3,2,0)(1,0,1)[12] : 1195.854
## ARIMA(3,2,0)(2,0,0)[12] : Inf
## ARIMA(3,2,1) : Inf
## ARIMA(3,2,1)(0,0,1)[12] : 1196.414
## ARIMA(3,2,1)(1,0,0)[12] : 1197.003
## ARIMA(3,2,2) : Inf
## ARIMA(4,2,0) : 1198.972
## ARIMA(4,2,0)(0,0,1)[12] : 1198.296
## ARIMA(4,2,0)(1,0,0)[12] : 1196.978
## ARIMA(4,2,1) : 1198.654
## ARIMA(5,2,0) : 1200.745
##
##
##
## Best model: ARIMA(2,2,0)(1,0,0)[12]auto_fit## Series: Electric_Vehicle_Population_Total_ts
## ARIMA(2,2,0)(1,0,0)[12]
##
## Coefficients:
## ar1 ar2 sar1
## -0.6136 -0.4319 0.3014
## s.e. 0.1113 0.1167 0.1322
##
## sigma^2 = 232459: log likelihood = -591.93
## AIC=1191.86 AICc=1192.41 BIC=1201.29attributes(auto_fit)## $names
## [1] "coef" "sigma2" "var.coef" "mask" "loglik" "aic"
## [7] "arma" "residuals" "call" "series" "code" "n.cond"
## [13] "nobs" "model" "bic" "aicc" "x" "fitted"
##
## $class
## [1] "forecast_ARIMA" "ARIMA" "Arima"Use arima and pass the values
The value (2,2,0) is the non-seasonal part of the ARIMA model. The value (1,0,0) is the seasonal part of the ARIMA model. Since this time series does not indicate any seasonality, the value (1,0,0) cannot be used and returns an error.
Electric_Vehicle_Population_Total_ts_arima <- arima(Electric_Vehicle_Population_Total_ts, order=c(2,2,0))
Electric_Vehicle_Population_Total_ts_arima##
## Call:
## arima(x = Electric_Vehicle_Population_Total_ts, order = c(2, 2, 0))
##
## Coefficients:
## ar1 ar2
## -0.6651 -0.5254
## s.e. 0.1017 0.1000
##
## sigma^2 estimated as 240715: log likelihood = -594.37, aic = 1194.74fit_Arima <- Arima(Electric_Vehicle_Population_Total_ts, order=c(2,2,0))
fit_Arima## Series: Electric_Vehicle_Population_Total_ts
## ARIMA(2,2,0)
##
## Coefficients:
## ar1 ar2
## -0.6651 -0.5254
## s.e. 0.1017 0.1000
##
## sigma^2 = 247062: log likelihood = -594.37
## AIC=1194.74 AICc=1195.06 BIC=1201.81Forecast with different confidence intervals and different forecasting time periods
The plot on the forecast shows the upward trend for the next 5 months or 12 months. It is a good model based on the plot.
forecast(fit_Arima, h=5, level = c(99.5))## Point Forecast Lo 99.5 Hi 99.5
## Sep 2023 155171.9 153776.6 156567.1
## Oct 2023 159624.9 157297.7 161952.1
## Nov 2023 164751.4 161539.4 167963.5
## Dec 2023 169554.5 165069.6 174039.4
## Jan 2024 174218.8 168384.7 180053.0forecast(fit_Arima, h=5, level = c(95))## Point Forecast Lo 95 Hi 95
## Sep 2023 155171.9 154197.6 156146.1
## Oct 2023 159624.9 158000.0 161249.8
## Nov 2023 164751.4 162508.7 166994.2
## Dec 2023 169554.5 166423.0 172686.0
## Jan 2024 174218.8 170145.2 178292.4forecast(fit_Arima, h=12, level = c(99.5))## Point Forecast Lo 99.5 Hi 99.5
## Sep 2023 155171.9 153776.6 156567.1
## Oct 2023 159624.9 157297.7 161952.1
## Nov 2023 164751.4 161539.4 167963.5
## Dec 2023 169554.5 165069.6 174039.4
## Jan 2024 174218.8 168384.7 180053.0
## Feb 2024 179145.4 171930.1 186360.7
## Mar 2024 183970.4 175201.8 192739.0
## Apr 2024 188725.2 178311.4 199138.9
## May 2024 193580.0 181458.7 205701.3
## Jun 2024 198405.2 184469.7 212340.7
## Jul 2024 203197.6 187364.5 219030.7
## Aug 2024 208027.4 190228.0 225826.7forecast(fit_Arima, h=12, level = c(95))## Point Forecast Lo 95 Hi 95
## Sep 2023 155171.9 154197.6 156146.1
## Oct 2023 159624.9 158000.0 161249.8
## Nov 2023 164751.4 162508.7 166994.2
## Dec 2023 169554.5 166423.0 172686.0
## Jan 2024 174218.8 170145.2 178292.4
## Feb 2024 179145.4 174107.4 184183.3
## Mar 2024 183970.4 177847.8 190093.0
## Apr 2024 188725.2 181454.0 195996.4
## May 2024 193580.0 185116.5 202043.5
## Jun 2024 198405.2 188675.0 208135.5
## Jul 2024 203197.6 192142.4 214252.8
## Aug 2024 208027.4 195599.3 220455.5forecast(fit_Arima, h=5)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Sep 2023 155171.9 154534.9 155808.9 154197.6 156146.1
## Oct 2023 159624.9 158562.4 160687.4 158000.0 161249.8
## Nov 2023 164751.4 163285.0 166217.9 162508.7 166994.2
## Dec 2023 169554.5 167506.9 171602.1 166423.0 172686.0
## Jan 2024 174218.8 171555.2 176882.4 170145.2 178292.4forecast(fit_Arima, h=12)## Point Forecast Lo 80 Hi 80 Lo 95 Hi 95
## Sep 2023 155171.9 154534.9 155808.9 154197.6 156146.1
## Oct 2023 159624.9 158562.4 160687.4 158000.0 161249.8
## Nov 2023 164751.4 163285.0 166217.9 162508.7 166994.2
## Dec 2023 169554.5 167506.9 171602.1 166423.0 172686.0
## Jan 2024 174218.8 171555.2 176882.4 170145.2 178292.4
## Feb 2024 179145.4 175851.2 182439.5 174107.4 184183.3
## Mar 2024 183970.4 179967.1 187973.7 177847.8 190093.0
## Apr 2024 188725.2 183970.8 193479.6 181454.0 195996.4
## May 2024 193580.0 188046.0 199114.0 185116.5 202043.5
## Jun 2024 198405.2 192043.0 204767.5 188675.0 208135.5
## Jul 2024 203197.6 195969.0 210426.2 192142.4 214252.8
## Aug 2024 208027.4 199901.1 216153.6 195599.3 220455.5plot(forecast(fit_Arima, h=5))
plot(forecast(fit_Arima, h=12))
Residual analysis for ARIMA forecast model
The Acf plot for residuals shows that the residuals are not significant hence random. This histogram of the residuals looks like a normal curve where the most values for redisuals are centered around 0.
Acf(fit_Arima$residuals)
Box.test(residuals(fit_Arima), lag=20, type="Ljung")##
## Box-Ljung test
##
## data: residuals(fit_Arima)
## X-squared = 11.641, df = 20, p-value = 0.9279plot.ts(residuals(fit_Arima))
hist(fit_Arima$residuals)
tsdiag(fit_Arima)
Accuracy for ARIMA forecast model
The MAPE of ARIMA forecast model is 0.5321711, which is pretty low. However, among all other forecasting models used to forecast this time series, Moving Average with order 3 still shows the lowest MAPE, 0.2403089. ARIMA is the second best forecasting model as compared to the remaining forecast modes.
accuracy (fit_Arima)## ME RMSE MAE MPE MAPE MASE
## Training set 113.645 484.4677 340.4416 0.1298684 0.5321711 0.01894405
## ACF1
## Training set -0.01457105